Next, we split the time interval \([t_{0}, t_{0}+ \Delta t]\)  into \([t_{0}, t_{0}+ \Delta t/2]\)  and \([t_{0}+\Delta t/2, t_{0} + \Delta t]\) . Then, we employ the conditional PDF that describes the intermediate position \(\mathbf {r}_{M}\)  at time \(t_{0}+\Delta t/2\) , given that the molecule was at time \(t_{0}\)  at \(\mathbf {r}_{0}\)  and ended up at time \(t_{0} + \Delta t\)  at \(\mathbf {r}\) . One can show that this conditional probability density is again a Gaussian with first moment given by \((\mathbf {r}_{0}+\mathbf {r})/2\)  and variance \(\sigma ^{2} \rightarrow \sigma ^{2}/2\)  as follows [1]}. Consider two independent Gaussian random variables \(\mathbf {X}\)  and \(\mathbf {Y}\) 
with variance \(\sigma ^{2}\)  and vanishing mean. To be definite, we consider the 3D case, but the described construction works equally well in 1 and 2D. The quantity we are interested in is the conditional PDF \(p(\mathbf {X}= \mathbf {x} \vert \mathbf {Z}:=\mathbf {X} + \mathbf {Y} = \mathbf {z})\) . In the context of a simulation, the variable \(\mathbf {Z}\)  describes the position update \(\mathbf {r}-\mathbf {r}_{0}\)  during a time step \(\Delta t\) , hence \(\mathbf {X}\)  can be interpreted to describe an intermediate position update \(\mathbf {r}_{M}-\mathbf {r}_{0}\)  during \(\Delta t/2\) . The conditional PDF can be obtained via
\(p(\mathbf {x}\vert \mathbf {z}) = \frac{p(\mathbf {x},\mathbf {z})}{p(\mathbf {z})},\) 
